Glueball spectrum based on a rigorous three-dimensional relativistic equation for two-gluon bound states. II. Calculation of the glueball spectrum

Abstract

In the preceding paper, a rigorous three-dimensional relativistic equation for two-gluon bound states was derived from the QCD with massive gluons and represented in the angular momentum representation. In order to apply this equation to calculate the glueball spectrum, in this paper, the equation is recast in an equivalent three-dimensional relativistic equation satisfied by the twogluon positive energy state amplitude. The interaction Hamiltonian in the equation is exactly derived and expressed as a perturbative series. The first term in the series describes the one-gluon exchange interaction which includes fully the retardation effect in it. This term plus the linear confining potential are chosen to be the interaction Hamiltonian and employed in the practical calculation. With the integrals containing three and four spherical Bessel functions in the QCD vertices being analytically calculated, the interaction Hamiltonian is given an explicit expression in the angular momentum representation. Numerically solving the relativistic equation with taking the contributions arising from the retardation effect and the longitudinal mode of gluon fields into account, a set of masses for the 0 ++ , 0 −+ , 1 ++ , 1 −+ , 2 ++ and 2 −+ glueball states are obtained and are in fairly good agreement with the predictions given by the lattice simulation In addition, some new glueball states are predicted.

Primordial non-Gaussianity and the CMB bispectrum

“…The integral in the brackets arises as a direct result of the delta function and analytic solutions have been found, [15][16][17]. These solutions demonstrate that the three lengths k 1 , k 2 , k 3 must be such as to be able to form a triangle, as expected from the delta function, which restricts us to the region of k-space highlighted in figure 1.…”

“…By looking at the expression for the reduced bispectrum (17), we can rearrange the terms as follows:…”

On a General Method for Resolving Integrals of Multiple Spherical Bessel Functions Against Power Laws into Distributions

“…We believe this could be done beginning from the integrals in the present work using a technique of [54], wherein one performs "parametric integration", to lower the power-law. Also of interest would be further investigating the reason for the singularity of some of these overlap integrals but not others; complementary perspectives may come from considering contour integration as an approach, as in [40], and also from analyzing the asymptotic behavior of the phase (e.g. as in equation ( 6…”

“…The recursion they find (their equation 47) is able to raise the order of one sBF in the integrand at a time. [40] computes three-SBF integrals with k 2 weight (their equation A18 and following), using contour integration and proving that the integral vanishes if the three free arguments cannot form a triangle. They also obtain a four-SBF integral with the same weight (their equation B11) and subject to the restrictions that its orders have an even sum; they show there are also several inequalities on the free arguments.…”

Elementary exact evaluation of infinite integrals of the product of several spherical Bessel functions, power and exponential

“…There are numerous articles devoted to analytical and numerical evaluation of infinite integrals, containing the maximum of three spherical Bessel functions. Such integrals are encountered in nuclear physics (Davies, 1988;Elbaz et al, 1974), in image processing (Baddour, 2010), in antenna design (Qia and others, 2007), particle physics (Anni, et al, 1974), astrophysics and in other fields of science and engineering (Jackson and Maximon, 1972;Chen and Su, 2004).…”